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IB Mathematics: Applications and Interpretation HL · 13.5: Integration · 13.5.4

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Definite integrals and area under a curve Problem set B

Write the definite integral first. Then evaluate it on the calculator and give 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Evaluate \(\int_{-3}^{2} 3x^4\,dx\).

  2. Evaluate \(\int_{1}^{4} (7-2t^3)\,dt\).

  3. Evaluate \(\int_{1}^{4} 6x^{-3}\,dx\).

  4. Evaluate \(\int_{0}^{2} (3x+2)^3\,dx\).

  5. The diagram shows the curve y = 15 − x². Find the area of the shaded region between x = 2 and x = 3.

    −11234551015xyy = 15 − x²

  6. The diagram shows the curve y = 2 + 8/x². Find the area of the shaded region between x = 2 and x = 5.

    12345612345678xyy = 2 + 8/x²

  7. The diagram shows the curve y = 8/x. Find the area of the shaded region between x = 2 and x = 7.

    1234567812345xyy = 8/x

  8. The diagram shows the curve \(y=\frac{x^2}{3}\) inside the rectangle with corners \((0,0)\), \((6,0)\), \((6,12)\) and \((0,12)\). Find the area of the shaded region above the curve.

    24682468101214xyy = x²/3y = 12

In context

  1. A pond lies between the x-axis and the curve \(y=6-0.5x^2\), from \(x=0\) to \(x=3\). Lengths are in metres. Find the area of the pond.

    −11234−11234567xyy = 6 − 0.5x²

  2. A news story is shared online at a rate of \(120e^{-0.15t}\) messages per day, where t is the time in days after it is posted. Find the total number of messages in the first 10 days.

  3. The cross-section of a greenhouse is the region between the x-axis (the ground) and the curve \(y=3-\frac{x^2}{3}\), where x and y are in metres. The greenhouse is 12 m long.

    −4−224−11234xyy = 3 − x²/3
    1. (a) Find the width of the greenhouse at ground level.
    2. (b) Find the area of the cross-section.
    3. (c) Find the volume of air inside the greenhouse.

Stretch

  1. The diagram shows the curve \(y=8-\dfrac{3}{x^2}\). It meets the x-axis at the point \((a,0)\).

    0.511.522.533.54−22468xyy = 8 − 3/x²
    1. (a) Find the exact value of a.
    2. (b) Find the area under the curve from x = a to x = 2.

jmaths.xyz · AI HL · 13.5.4 Definite integrals and area under a curve